IMO Shortlist 2004 problem C1

There are students at an university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of societies. Suppose that the following conditions hold:

i.) Each pair of students are in exactly one club.

ii.) For each student and each society, the student is in exactly one club of the society.

iii.) Each club has an odd number of students. In addition, a club with students ( is a positive integer) is in exactly societies.

Find all possible values of . RemarkIn IMOTC 2005, it is given that .

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There are $10001$ students at an university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of $k$ societies. Suppose that the following conditions hold:
i.) Each pair of students are in exactly one club.
ii.) For each student and each society, the student is in exactly one club of the society.
iii.) Each club has an odd number of students. In addition, a club with ${2m+1}$ students ($m$ is a positive integer) is
in exactly $m$ societies.
Find all possible values of $k$.
RemarkIn IMOTC 2005, it is given that $m=2005$.